Roadmap

• Descriptive evidence about firm death and dividends

• A toy model that explains some of this evidence

• (If time permits) a connection to the equity volatility puzzle

Defining Firm Death

How I am working with firm death:

Given the pricing kernel, \(M\), and payoffs, \(X\), when would

\[ P = E\left(MX\right) = 0 \]

given perfect knowledge of firm cashflows.

The data

• I use all firms that have ever paid a quarterly dividend in CRSP, since 1926. The sample choice matters little.
• Firms “die” when they never pay a cashflow again (this is a minimum of 3 years)
• In an ideal world I would include buybacks; this was harder than I thought
• I tried to include other kinds of returns of capital and found only small differences

How much does death matter?

Dividend Cuts and the Hazard Rate

Model

Three periods, \(t \in \{0, 1, 2\}\)

Two types, \(v \in \{L, H\}\).

Let \(s_{v, t}\) denote the share of firms of type \(v\) at time \(t\).

Firms have earnings, \(e_t\) at time \(t \in \{1, 2\}\), with

\[ e_t = \begin{cases} \bar{e} \ \text{with probability } \lambda_v \\ \underline{e} \ \text{with probability } (1 - \lambda_v) \end{cases} \]

Model

To alleviate incentive problems, managers are paid a portion of the stock price at periods \(t \in \{0, 1\}\), where

\[ S_0 = E^M_t\left(d_1 + d_2\right)\\ S_1 = E^M_t\left(d_2\right)\\ S_2 = 0 \]

Managers have wage \(w_t = \alpha S_t\) for some \(\alpha \in (0, 1)\). Managers know their types, but the market does not.

Model

In periods \(t \in \{0, 1\}\) managers commit to a dividend policy which is a function \(d_t: e_{t+1} \to \mathbb{R}\).

This is subject to the constraint

\[ \Delta c_t + d_t \leq e_t \]

with the additional stipulation that if \(c_1 = 0\) the firm dies in that period and cannot receive earnings in period 2.

Note that in any equilibrium, managers commit to liquidating the firm at the end of period 2.

Model

Model

There are two equilibria of interest here.

1. The Separating Equilibrium
2. The Dividend Irrelevance / MM Equilibrium

Separating Equilibrium, Strategies

Strategy profiles:

High types, \(v = {H}\)

\[d_{1,H} = \begin{cases} \underline{e} \text{ when } e_1 = \underline{e} \\ 0 \lt d_1 \lt \bar{e} \text{ otherwise} \end{cases}\]

Types \(H\) commit to ritual suicide in period one in order to signal their type distribution of firms in period two.

Separating Equilibrium, Strategies

Low types, \(v = {L}\)

\[d_{1,L} = 0 \leq d_1 \lt e_1\]

Since there is no discounting, as long as they survive to period 2 their valuation is the same.

Separating Equilibrium, Conditions

High types don’t deviate if

\[S_{1, H} + \lambda_H S_{1, H} - 2S_{1, L} > 0\]

Low types don’t deviate if

\[2S_{1, L} - S_{1, H} - \lambda_L S_{1, H} > 0\]

Equity Volatility - Individual Firms

Let’s start with Shiller’s constant discount rate model:

\[ S_{i,t} = E^M\left(\sum_{s=t}^\infty \frac{d_sP(\text{alive})}{(1 + r)^t} \right) \]

Suppose we have a survival rate, \(\lambda\), where dividends contain information about \(\lambda\). Then

\[ S_{i,t} = E^M\left(\sum_{s=t}^\infty \frac{d_s \lambda(d_{i, 0:t})^t}{(1 + r)^t} \right) \]

Equity Volatility - Individual Firms

If we assume a constant future dividend (just for simplicity), we get

\[ S_{i, t} = \frac{d_{i,t}}{r - (\lambda(d_{i,0:t}) - 1)} = \frac{d_{i,t}}{r + (1 - \lambda(d_{i,0:t}))} \]

which implies that changes in the hazard rate look like changes in the discount rate.

Equity Volatility - Aggregating

Conclusions / Next Steps

• At the individual firm level, dividends are very informative about the path of future cashflows

• This pattern can be rationalized with a signaling model

• It seems like this effect is not large enough to generate the variation in equity prices that we would need

• Next steps would be to make the model dynamic / have a continuum of firms / have a continuum of payoffs (maybe in continuous time?)